where \({\bf {g}}_{m_2}^T\in {\mathbb {C}}^{1\times N}\)  is the \(m_2\) th row vector of the uplink channel \(\bf {G}\) , \({\bf {g}}_{m_1}\in {\mathbb {C}}^{N\times 1}\)  is the \(m_1\) th row vector of the downlink channel \({\bf {G}}^T\) , \({\mathbf {\theta }}_t\)  is the reflecting vector at the RIS in the \(t\) th sub-frame, \(z_{m_1,m_2}\)  is the self-interference after mitigation from the \(m_1\) th antenna to the \(m_2\) th antenna of the BS, and \(w_{m_1,m_2,t}\)  is the received noise at the \(m_2\) th antenna of the BS. After \(N+1\)  sub-frames, all received pilot signals \(\lbrace y_{m_1,m_2,t}|1\le m_1,m_2 \le N, m_1\ne m_2, 1\le t \le N+1\rbrace \)  can be obtained, which consist of \(MN\)  unknown variables, i.e., \(MN\)  elements of \(\bf {G}\) . Then, based on all received pilots, each element of \(\bf {G}\)  can be alternately estimated in an iterative manner by utilizing the coordinate descent algorithm [1]}.
